# Basic Laplace Transform of a 1st order Differential Equation

• Oct 12th 2011, 01:19 PM
craig
Basic Laplace Transform of a 1st order Differential Equation
Just wondering if someone could have a quick look over what I've done so far:

Assuming zero initial conditions, solve the following DE:

$\dot{x} + x = 2$

The Laplace Transform gives:

$sX(s) - x_0 + \frac{1}{s^2} = \frac{2}{s}$

Rearranging and using the fact that $x_0 =0$ gives us

$X(s) = \frac{2}{s^2} - \frac{1}{s^3}$

$X(s) = 2\frac{1!}{s^{1+1}} - \frac{1}{2} \frac{2!}{s^{2+1}}$

So $x(s) = 2t - \frac{t^2}{2}$?
• Oct 12th 2011, 02:09 PM
Jester
Re: Basic Laplace Transform of a 1st order Differential Equation
After taking the Laplace transform, where did the $\frac{1}{s^2}$ come from?
• Oct 12th 2011, 11:11 PM
craig
Re: Basic Laplace Transform of a 1st order Differential Equation
Quote:

Originally Posted by Danny
After taking the Laplace transform, where did the $\frac{1}{s^2}$ come from?

I took the Laplace transform of $x$, is this not right?
• Oct 12th 2011, 11:40 PM
FernandoRevilla
Re: Basic Laplace Transform of a 1st order Differential Equation
Quote:

Originally Posted by craig
I took the Laplace transform of $x$, is this not right?

$x$ is a dependent variable ( $x=x(t)$ ), so you'll obtain

$s\mathcal{L}\{x\}-x_0+\mathcal{L}\{x\}=\frac{2}{s}\Rightarrow (s+1)\mathcal{L}\{x\}=x_0+\frac{2}{s}\Rightarrow \ldots$
• Oct 13th 2011, 12:30 AM
craig
Re: Basic Laplace Transform of a 1st order Differential Equation
Quote:

Originally Posted by FernandoRevilla
$x$ is a dependent variable ( $x=x(t)$ ), so you'll obtain

$s\mathcal{L}\{x\}-x_0+\mathcal{L}\{x\}=\frac{2}{s}\Rightarrow (s+1)\mathcal{L}\{x\}=x_0+\frac{2}{s}\Rightarrow \ldots$

So given that $x_0 = 0$, does that mean that we just have $\mathcal{L}\{x\}=\frac{2}{s(s+1)}$.

Which I presume we express in partial fractions, and then solve for $x(t)$?
• Oct 13th 2011, 12:38 AM
craig
Re: Basic Laplace Transform of a 1st order Differential Equation
So for $\mathcal{L}\{x\}=\frac{2}{s(s+1)} = \frac{2}{s} - \frac{2}{s+1}$ we have:

$\mathcal{L}\{x\}= 2\frac{1}{s} - 2\frac{1}{s-(-1)}$

So $x(t) = 2(1) - 2 (e^{-1t}) = 2(1-e^{-t)$ ?

Thanks again for the replies.
• Oct 13th 2011, 01:25 AM
FernandoRevilla
Re: Basic Laplace Transform of a 1st order Differential Equation
Quote:

Originally Posted by craig
So $x(t) = 2(1) - 2 (e^{-1t}) = 2(1-e^{-t)$ ?

Right.
• Oct 13th 2011, 01:43 AM
Ackbeet
Re: Basic Laplace Transform of a 1st order Differential Equation
Quote:

Originally Posted by craig
So for $\mathcal{L}\{x\}=\frac{2}{s(s+1)} = \frac{2}{s} - \frac{2}{s+1}$ we have:

$\mathcal{L}\{x\}= 2\frac{1}{s} - 2\frac{1}{s-(-1)}$

So $x(t) = 2(1) - 2 (e^{-1t}) = 2(1-e^{-t)$ ?

Thanks again for the replies.

Technically, since you're working with the one-sided LT (I assume), you have unit step functions multiplying everything. That is,

$x(t)=2u(t)(1-e^{-t}),$

and the solution is valid for non-negative t's.
• Oct 13th 2011, 02:05 AM
FernandoRevilla
Re: Basic Laplace Transform of a 1st order Differential Equation
Quote:

Originally Posted by Ackbeet
and the solution is valid for non-negative t's.

Also considering the theorem about the necessary form of the solutions for the equation $x^{(n)}+a_{n-1}x^{(n-1)}+\ldots +a_1x'+a_0x=b(t)$ .
• Oct 13th 2011, 02:18 AM
craig
Re: Basic Laplace Transform of a 1st order Differential Equation
Quote:

Originally Posted by FernandoRevilla
Also considering the theorem about the necessary form of the solutions for the equation $x^{(n)}+a_{n-1}x^{(n-1)}+\ldots +a_1x'+a_0x=b(t)$ .

Sorry would you mind explaining what you meant by this? I understood the first part about the Unit-Step function, just not sure what you meant here?

Thankyou again!
• Oct 13th 2011, 04:08 AM
FernandoRevilla
Re: Basic Laplace Transform of a 1st order Differential Equation
Quote:

Originally Posted by craig
Sorry would you mind explaining what you meant by this?

The Laplace transform of $x:[0,+\infty)\to\mathbb{R}$ is defined by $\mathcal{L}\{x(t)\}=\int_0^{+\infty}e^{-st}x(t)\;dt$ , so a priori we find solutions valid for $t\geq 0$. Your equation has (by a well known theorem) a determined form for its unique solution $x(t)=k+Ce^{at}$ valid for all $t\in\mathbb{R}$ so, necessarily is valid the solution found by LT for all $t\in\mathbb{R}$ .
• Oct 13th 2011, 04:32 AM
Ackbeet
Re: Basic Laplace Transform of a 1st order Differential Equation
Quote:

Originally Posted by FernandoRevilla
The Laplace transform of $x:[0,+\infty)\to\mathbb{R}$ is defined by $\mathcal{L}\{x(t)\}=\int_0^{+\infty}e^{-st}x(t)\;dt$ , so a priori we find solutions valid for $t\geq 0$. Your equation has (by a well known theorem) a determined form for its unique solution $x(t)=k+Ce^{at}$ valid for all $t\in\mathbb{R}$ so, necessarily is valid the solution found by LT for all $t\in\mathbb{R}$ .

Agreed. You could simply say that you extend the solution found by the LT method to the entire real line. The result of the inverse LT does have the unit step functions in it, and, in fact, does not satisfy the DE for negative t's. However, with the theorem you have invoked, you can extend the solution by eliminating the unit step function.
• Oct 13th 2011, 11:57 PM
craig
Re: Basic Laplace Transform of a 1st order Differential Equation
Thankyou both! I don't think we've covered this in lectures yet but I think it makes sense.